# Computer Algebra for Special Functions Inequalities [P22748-N18]

### Project Lead

### Project Duration

01/10/2010 - 30/09/2013## Partners

### The Austrian Science Fund (FWF)

## Publications

### 2018

### Algorithmic Arithmetics with DD-Finite Functions

#### Jiménez-Pastor Antonio, Pillwein Veronika

In: Proceedings of the 2018 ACM on International Symposium on Symbolic and Algebraic Computation, Arreche Carlos (ed.), ISSAC '18 , pp. 231-237. 2018. ACM, New York, NY, USA, ISBN 978-1-4503-5550-6. [url]@

author = {Jiménez-Pastor Antonio and Pillwein Veronika},

title = {{Algorithmic Arithmetics with DD-Finite Functions}},

booktitle = {{Proceedings of the 2018 ACM on International Symposium on Symbolic and Algebraic Computation}},

language = {english},

series = {ISSAC '18},

pages = {231--237},

publisher = {ACM},

address = {New York, NY, USA},

isbn_issn = {ISBN 978-1-4503-5550-6},

year = {2018},

editor = {Arreche Carlos},

refereed = {yes},

keywords = {algorithms, closure properties, holonomic functions},

length = {7},

url = {http://doi.acm.org/10.1145/3208976.3209009}

}

**inproceedings**{RISC5730,author = {Jiménez-Pastor Antonio and Pillwein Veronika},

title = {{Algorithmic Arithmetics with DD-Finite Functions}},

booktitle = {{Proceedings of the 2018 ACM on International Symposium on Symbolic and Algebraic Computation}},

language = {english},

series = {ISSAC '18},

pages = {231--237},

publisher = {ACM},

address = {New York, NY, USA},

isbn_issn = {ISBN 978-1-4503-5550-6},

year = {2018},

editor = {Arreche Carlos},

refereed = {yes},

keywords = {algorithms, closure properties, holonomic functions},

length = {7},

url = {http://doi.acm.org/10.1145/3208976.3209009}

}

### A Computable Extension for Holonomic Functions: DD-Finite Functions

#### Jiménez-Pastor Antonio, Pillwein Veronika

Journal of Symbolic Computation, pp. -. 2018. ISSN 0747-7171. accepted.@

author = {Jiménez-Pastor Antonio and Pillwein Veronika},

title = {{A Computable Extension for Holonomic Functions: DD-Finite Functions}},

language = {english},

journal = {Journal of Symbolic Computation},

pages = {--},

isbn_issn = {ISSN 0747-7171},

year = {2018},

note = {accepted},

refereed = {yes},

length = {0}

}

**article**{RISC5731,author = {Jiménez-Pastor Antonio and Pillwein Veronika},

title = {{A Computable Extension for Holonomic Functions: DD-Finite Functions}},

language = {english},

journal = {Journal of Symbolic Computation},

pages = {--},

isbn_issn = {ISSN 0747-7171},

year = {2018},

note = {accepted},

refereed = {yes},

length = {0}

}

### 2016

### A hypergeometric inequality

#### A. Dixit, V.H. Moll, V. Pillwein

Annals of Combinatorics 20(1), pp. 65-72. 2016. 0218-0006. [pdf]@

author = {A. Dixit and V.H. Moll and V. Pillwein},

title = {{A hypergeometric inequality}},

language = {english},

journal = {Annals of Combinatorics},

volume = {20},

number = {1},

pages = {65--72},

isbn_issn = {0218-0006},

year = {2016},

refereed = {yes},

length = {8}

}

**article**{RISC4965,author = {A. Dixit and V.H. Moll and V. Pillwein},

title = {{A hypergeometric inequality}},

language = {english},

journal = {Annals of Combinatorics},

volume = {20},

number = {1},

pages = {65--72},

isbn_issn = {0218-0006},

year = {2016},

refereed = {yes},

length = {8}

}

### 2015

### Polyhedral Omega: A New Algorithm for Solving Linear Diophantine Systems

#### Felix Breuer, Zafeirakis Zafeirakopoulos

Technical report no. 15-09 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Schloss Hagenberg, 4232 Hagenberg, Austria. January 2015. [pdf]@

author = {Felix Breuer and Zafeirakis Zafeirakopoulos},

title = {{Polyhedral Omega: A New Algorithm for Solving Linear Diophantine Systems}},

language = {english},

abstract = {Polyhedral Omega is a new algorithm for solving linear Diophantine systems (LDS), i.e., for computing a multivariate rational function representation of the set of all non-negative integer solutions to a system of linear equations and inequalities. Polyhedral Omega combines methods from partition analysis with methods from polyhedral geometry. In particular, we combine MacMahon's iterative approach based on the Omega operator and explicit formulas for its evaluation with geometric tools such as Brion decompositions and Barvinok's short rational function representations. In this way, we connect two recent branches of research that have so far remained separate, unified by the concept of symbolic cones which we introduce. The resulting LDS solver Polyhedral Omega is significantly faster than previous solvers based on partition analysis and it is competitive with state-of-the-art LDS solvers based on geometric methods. Most importantly, this synthesis of ideas makes Polyhedral Omega the simplest algorithm for solving linear Diophantine systems available to date. Moreover, we provide an illustrated geometric interpretation of partition analysis, with the aim of making ideas from both areas accessible to readers from a wide range of backgrounds.},

number = {15-09},

year = {2015},

month = {January},

keywords = {Linear Diophantine system, linear inequality system, integer solutions, partition analysis, partition theory, polyhedral geometry, rational function, symbolic cone, generating function, implementation},

length = {49},

type = {RISC Report Series},

institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},

address = {Schloss Hagenberg, 4232 Hagenberg, Austria}

}

**techreport**{RISC5153,author = {Felix Breuer and Zafeirakis Zafeirakopoulos},

title = {{Polyhedral Omega: A New Algorithm for Solving Linear Diophantine Systems}},

language = {english},

abstract = {Polyhedral Omega is a new algorithm for solving linear Diophantine systems (LDS), i.e., for computing a multivariate rational function representation of the set of all non-negative integer solutions to a system of linear equations and inequalities. Polyhedral Omega combines methods from partition analysis with methods from polyhedral geometry. In particular, we combine MacMahon's iterative approach based on the Omega operator and explicit formulas for its evaluation with geometric tools such as Brion decompositions and Barvinok's short rational function representations. In this way, we connect two recent branches of research that have so far remained separate, unified by the concept of symbolic cones which we introduce. The resulting LDS solver Polyhedral Omega is significantly faster than previous solvers based on partition analysis and it is competitive with state-of-the-art LDS solvers based on geometric methods. Most importantly, this synthesis of ideas makes Polyhedral Omega the simplest algorithm for solving linear Diophantine systems available to date. Moreover, we provide an illustrated geometric interpretation of partition analysis, with the aim of making ideas from both areas accessible to readers from a wide range of backgrounds.},

number = {15-09},

year = {2015},

month = {January},

keywords = {Linear Diophantine system, linear inequality system, integer solutions, partition analysis, partition theory, polyhedral geometry, rational function, symbolic cone, generating function, implementation},

length = {49},

type = {RISC Report Series},

institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},

address = {Schloss Hagenberg, 4232 Hagenberg, Austria}

}

### An extension of Turan's inequality for ultraspherical polynomials

#### G. Nikolov, V. Pillwein

Mathematical Inequalities & Applications 18(1), pp. 321-335. 2015. [pdf]@

author = {G. Nikolov and V. Pillwein},

title = {{An extension of Turan's inequality for ultraspherical polynomials}},

language = {english},

journal = {Mathematical Inequalities & Applications},

volume = {18},

number = {1},

pages = {321--335},

isbn_issn = {?},

year = {2015},

refereed = {yes},

length = {15}

}

**article**{RISC4953,author = {G. Nikolov and V. Pillwein},

title = {{An extension of Turan's inequality for ultraspherical polynomials}},

language = {english},

journal = {Mathematical Inequalities & Applications},

volume = {18},

number = {1},

pages = {321--335},

isbn_issn = {?},

year = {2015},

refereed = {yes},

length = {15}

}

### Symbolic Computation and Finite Element Methods.

#### V. Pillwein

In: CASC 2015, V.P. Gerdt, W. Koepf, W.M. Seiler, and E.V. Vorozhtsov (ed.), LNCS 9301, pp. 374-388. 2015. Springer-Verlag Berlin Heidelberg, 0302-9743. [pdf]@

author = {V. Pillwein},

title = {{Symbolic Computation and Finite Element Methods.}},

booktitle = {{CASC 2015}},

language = {english},

series = {LNCS},

volume = {9301},

pages = {374--388},

publisher = {Springer-Verlag Berlin Heidelberg},

isbn_issn = {0302-9743},

year = {2015},

editor = {V.P. Gerdt and W. Koepf and W.M. Seiler and and E.V. Vorozhtsov},

refereed = {no},

length = {15}

}

**inproceedings**{RISC5182,author = {V. Pillwein},

title = {{Symbolic Computation and Finite Element Methods.}},

booktitle = {{CASC 2015}},

language = {english},

series = {LNCS},

volume = {9301},

pages = {374--388},

publisher = {Springer-Verlag Berlin Heidelberg},

isbn_issn = {0302-9743},

year = {2015},

editor = {V.P. Gerdt and W. Koepf and W.M. Seiler and and E.V. Vorozhtsov},

refereed = {no},

length = {15}

}

### Efficient Computation of Multiplicity and Directional Multiplicity of an Isolated Point

#### Angelos Mantzaflaris, Hamid Rahkooy, Zafeirakis Zafeirakopoulos

Technical report no. 15-08 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Schloss Hagenberg, 4232 Hagenberg, Austria. 2015. [pdf]@

author = { Angelos Mantzaflaris and Hamid Rahkooy and Zafeirakis Zafeirakopoulos},

title = {{Efficient Computation of Multiplicity and Directional Multiplicity of an Isolated Point}},

language = {english},

number = {15-08},

year = {2015},

length = {25},

type = {RISC Report Series},

institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},

address = {Schloss Hagenberg, 4232 Hagenberg, Austria}

}

**techreport**{RISC5152,author = { Angelos Mantzaflaris and Hamid Rahkooy and Zafeirakis Zafeirakopoulos},

title = {{Efficient Computation of Multiplicity and Directional Multiplicity of an Isolated Point}},

language = {english},

number = {15-08},

year = {2015},

length = {25},

type = {RISC Report Series},

institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},

address = {Schloss Hagenberg, 4232 Hagenberg, Austria}

}

### 2014

### Comparison between binary and decimal floating-point numbers

#### Nicolas Brisebarre, Christoph Lauter, Marc Mezzarobba, Jean-Michel Muller

HAL. Technical report no. hal-01021928, 2014. [url]@

author = {Nicolas Brisebarre and Christoph Lauter and Marc Mezzarobba and Jean-Michel Muller},

title = {{Comparison between binary and decimal floating-point numbers}},

language = {english},

number = {hal-01021928},

year = {2014},

institution = {HAL},

length = {13},

url = {http://hal.archives-ouvertes.fr/hal-01021928/}

}

**techreport**{RISC5020,author = {Nicolas Brisebarre and Christoph Lauter and Marc Mezzarobba and Jean-Michel Muller},

title = {{Comparison between binary and decimal floating-point numbers}},

language = {english},

number = {hal-01021928},

year = {2014},

institution = {HAL},

length = {13},

url = {http://hal.archives-ouvertes.fr/hal-01021928/}

}

### Rigorous Uniform Approximation of D-finite Functions Using Chebyshev Expansions

#### Alexandre Benoit, Mioara Joldeș, Marc Mezzarobba

HAL. Technical report no. hal-01022420, 2014. [url]@

author = {Alexandre Benoit and Mioara Joldeș and Marc Mezzarobba},

title = {{Rigorous Uniform Approximation of D-finite Functions Using Chebyshev Expansions}},

language = {english},

number = {hal-01022420},

year = {2014},

institution = {HAL},

length = {39},

url = {http://hal.archives-ouvertes.fr/hal-01022420/}

}

**techreport**{RISC5021,author = {Alexandre Benoit and Mioara Joldeș and Marc Mezzarobba},

title = {{Rigorous Uniform Approximation of D-finite Functions Using Chebyshev Expansions}},

language = {english},

number = {hal-01022420},

year = {2014},

institution = {HAL},

length = {39},

url = {http://hal.archives-ouvertes.fr/hal-01022420/}

}

### A note on uniform approximation of functions having a double pole

#### I. Moale, V. Pillwein

LMS J. Comput. Math. 17(1), pp. 233-244. 2014. [pdf]@

author = {I. Moale and V. Pillwein},

title = {{A note on uniform approximation of functions having a double pole}},

language = {english},

journal = {LMS J. Comput. Math.},

volume = {17},

number = {1},

pages = {233--244},

isbn_issn = {?},

year = {2014},

refereed = {yes},

length = {12}

}

**article**{RISC4908,author = {I. Moale and V. Pillwein},

title = {{A note on uniform approximation of functions having a double pole}},

language = {english},

journal = {LMS J. Comput. Math.},

volume = {17},

number = {1},

pages = {233--244},

isbn_issn = {?},

year = {2014},

refereed = {yes},

length = {12}

}

### A local Fourier convergence analysis of a multigrid method using symbolic computation

#### V. Pillwein, S. Takacs

Journal of Symbolic Computation 63, pp. 1-20. 2014. ISSN:0747-7171 . [pdf]@

author = {V. Pillwein and S. Takacs},

title = {{A local Fourier convergence analysis of a multigrid method using symbolic computation}},

language = {english},

journal = {Journal of Symbolic Computation},

volume = {63},

pages = {1--20},

isbn_issn = {ISSN:0747-7171 },

year = {2014},

refereed = {yes},

length = {20}

}

**article**{RISC4907,author = {V. Pillwein and S. Takacs},

title = {{A local Fourier convergence analysis of a multigrid method using symbolic computation}},

language = {english},

journal = {Journal of Symbolic Computation},

volume = {63},

pages = {1--20},

isbn_issn = {ISSN:0747-7171 },

year = {2014},

refereed = {yes},

length = {20}

}

### 2013

### Harmonic interpolation based on Radon projections along the sides of regular polygons

#### Irina Georgieva, Clemens Hofreither, Christoph Koutschan, Veronika Pillwein, Thotsaporn Thanatipanonda

Central European Journal of Mathematics 11(4), pp. 609-620. 2013. ISSN 1895-1074. [pdf]@

author = {Irina Georgieva and Clemens Hofreither and Christoph Koutschan and Veronika Pillwein and Thotsaporn Thanatipanonda},

title = {{Harmonic interpolation based on Radon projections along the sides of regular polygons}},

language = {english},

abstract = {Given information about a harmonic function in two variables, consisting of a finitenumber of values of its Radon projections, i.e., integrals along some chords of the unitcircle, we study the problem of interpolating these data by a harmonic polynomial.With the help of symbolic summation techniques we show that this interpolationproblem has a unique solution in the case when the chords form a regular polygon.Numerical experiments for this and more general cases are presented.},

journal = {Central European Journal of Mathematics},

volume = {11},

number = {4},

pages = {609--620},

isbn_issn = {ISSN 1895-1074},

year = {2013},

refereed = {yes},

length = {12}

}

**article**{RISC4655,author = {Irina Georgieva and Clemens Hofreither and Christoph Koutschan and Veronika Pillwein and Thotsaporn Thanatipanonda},

title = {{Harmonic interpolation based on Radon projections along the sides of regular polygons}},

language = {english},

abstract = {Given information about a harmonic function in two variables, consisting of a finitenumber of values of its Radon projections, i.e., integrals along some chords of the unitcircle, we study the problem of interpolating these data by a harmonic polynomial.With the help of symbolic summation techniques we show that this interpolationproblem has a unique solution in the case when the chords form a regular polygon.Numerical experiments for this and more general cases are presented.},

journal = {Central European Journal of Mathematics},

volume = {11},

number = {4},

pages = {609--620},

isbn_issn = {ISSN 1895-1074},

year = {2013},

refereed = {yes},

length = {12}

}

### Sparsity optimized high order finite element functions for H(curl) on tetrahedra

#### S. Beuchler and V. Pillwein and S. Zaglmayr

Advances in Applied Mathematics 50, pp. 749-769. 2013. [url] [pdf]@

author = {S. Beuchler and V. Pillwein and S. Zaglmayr},

title = {{Sparsity optimized high order finite element functions for H(curl) on tetrahedra}},

language = {english},

journal = {Advances in Applied Mathematics},

volume = {50},

pages = {749--769},

isbn_issn = {?},

year = {2013},

refereed = {yes},

length = {20},

url = {http://dx.doi.org/10.1016/j.aam.2012.11.004}

}

**article**{RISC4663,author = {S. Beuchler and V. Pillwein and S. Zaglmayr},

title = {{Sparsity optimized high order finite element functions for H(curl) on tetrahedra}},

language = {english},

journal = {Advances in Applied Mathematics},

volume = {50},

pages = {749--769},

isbn_issn = {?},

year = {2013},

refereed = {yes},

length = {20},

url = {http://dx.doi.org/10.1016/j.aam.2012.11.004}

}

### Termination Conditions for Positivity Proving Procedures

#### V. Pillwein

In: Proceedings of ISSAC'13, M. Kauers (ed.), pp. 315-322. 2013. isbn 978-1-4503-2059-7/13/06. [pdf]@

author = {V. Pillwein},

title = {{Termination Conditions for Positivity Proving Procedures}},

booktitle = {{Proceedings of ISSAC'13}},

language = {english},

abstract = { Proving positivity of a sequence given by a linear recurrence with polynomial coefficients (P-finite recurrence) is a non-trivial task for both humans and computers. Algorithms dealing with this task are rare or non-existent. One method that was introduced in the last decade by Gerhold and Kauers succeeds on many examples, but termination of this procedure has been proven so far only up to order three for special cases. Here we present an analysis that extends the previously known termination results on recurrences of order three, and also provides termination conditions for recurrences of higher order.},

pages = {315--322},

isbn_issn = {isbn 978-1-4503-2059-7/13/06},

year = {2013},

editor = {M. Kauers},

refereed = {yes},

length = {7}

}

**inproceedings**{RISC4714,author = {V. Pillwein},

title = {{Termination Conditions for Positivity Proving Procedures}},

booktitle = {{Proceedings of ISSAC'13}},

language = {english},

abstract = { Proving positivity of a sequence given by a linear recurrence with polynomial coefficients (P-finite recurrence) is a non-trivial task for both humans and computers. Algorithms dealing with this task are rare or non-existent. One method that was introduced in the last decade by Gerhold and Kauers succeeds on many examples, but termination of this procedure has been proven so far only up to order three for special cases. Here we present an analysis that extends the previously known termination results on recurrences of order three, and also provides termination conditions for recurrences of higher order.},

pages = {315--322},

isbn_issn = {isbn 978-1-4503-2059-7/13/06},

year = {2013},

editor = {M. Kauers},

refereed = {yes},

length = {7}

}

### On Computing Elimination Ideals Using Resultants with Applications to Groebner Bases

#### Hamid Rahkooy, Zafeirakis Zafeirakopoulos

Research Institute for Symbolic Computations, Doctoral College Computational Mathematics. Technical report, 2013. [url]@

author = {Hamid Rahkooy and Zafeirakis Zafeirakopoulos},

title = {{On Computing Elimination Ideals Using Resultants with Applications to Groebner Bases}},

language = {english},

year = {2013},

institution = {Research Institute for Symbolic Computations, Doctoral College Computational Mathematics},

length = {23},

url = {https://www.dk-compmath.jku.at/publications/dk-reports/2013-05-21/view}

}

**techreport**{RISC4731,author = {Hamid Rahkooy and Zafeirakis Zafeirakopoulos},

title = {{On Computing Elimination Ideals Using Resultants with Applications to Groebner Bases}},

language = {english},

year = {2013},

institution = {Research Institute for Symbolic Computations, Doctoral College Computational Mathematics},

length = {23},

url = {https://www.dk-compmath.jku.at/publications/dk-reports/2013-05-21/view}

}

### A Groebner Bases Method for Complementary Sequences

#### Koukouvinos Christos, Simos Dimitris E, Zafeirakopoulos Zafeirakis

In: Proceedings of Applications of Computer Algebra, Jose Luis Galan Garcia, Gabriel Aguilera Venegas, Pedro Rodirguez Cielos (ed.), pp. 255-260. 2013. 84-616-4565-0.@

author = {Koukouvinos Christos and Simos Dimitris E and Zafeirakopoulos Zafeirakis},

title = {{A Groebner Bases Method for Complementary Sequences}},

booktitle = {{Proceedings of Applications of Computer Algebra}},

language = {english},

pages = {255--260},

isbn_issn = {84-616-4565-0},

year = {2013},

editor = {Jose Luis Galan Garcia and Gabriel Aguilera Venegas and Pedro Rodirguez Cielos},

refereed = {yes},

length = {5}

}

**inproceedings**{RISC4823,author = {Koukouvinos Christos and Simos Dimitris E and Zafeirakopoulos Zafeirakis},

title = {{A Groebner Bases Method for Complementary Sequences}},

booktitle = {{Proceedings of Applications of Computer Algebra}},

language = {english},

pages = {255--260},

isbn_issn = {84-616-4565-0},

year = {2013},

editor = {Jose Luis Galan Garcia and Gabriel Aguilera Venegas and Pedro Rodirguez Cielos},

refereed = {yes},

length = {5}

}

### 2012

### Closed form solutions of linear difference equations in terms of symmetric products

#### Y. Cha, V. Pillwein

ACM Communications in Computer Algebra 46(3-4), pp. 80-81. Sep 2012. 0.@

author = {Y. Cha and V. Pillwein},

title = {{Closed form solutions of linear difference equations in terms of symmetric products}},

language = {english},

journal = {ACM Communications in Computer Algebra},

volume = {46},

number = {3-4},

pages = {80--81},

isbn_issn = {0},

year = {2012},

month = {Sep},

refereed = {yes},

length = {2}

}

**article**{RISC5616,author = {Y. Cha and V. Pillwein},

title = {{Closed form solutions of linear difference equations in terms of symmetric products}},

language = {english},

journal = {ACM Communications in Computer Algebra},

volume = {46},

number = {3-4},

pages = {80--81},

isbn_issn = {0},

year = {2012},

month = {Sep},

refereed = {yes},

length = {2}

}

### A local Fourier convergence analysis of a multigrid method using symbolic computation

#### V. Pillwein, S. Takacs

DK Computational Mathematics. Technical report no. DK Report 2012-04, April 2012. [pdf]@

author = {V. Pillwein and S. Takacs},

title = {{A local Fourier convergence analysis of a multigrid method using symbolic computation}},

language = {english},

number = {DK Report 2012-04},

year = {2012},

month = {April},

institution = {DK Computational Mathematics},

length = {20}

}

**techreport**{RISC4526,author = {V. Pillwein and S. Takacs},

title = {{A local Fourier convergence analysis of a multigrid method using symbolic computation}},

language = {english},

number = {DK Report 2012-04},

year = {2012},

month = {April},

institution = {DK Computational Mathematics},

length = {20}

}

### 2011

### Dominance in the Family of Sugeno-Weber t-norms

#### Manuel Kauers, Veronika Pillwein, Susanne Saminger-Platz

Fuzzy Sets and Systems 181(1), pp. 74-87. October 2011. ISSN 0165-0114. [ps] [pdf]@

author = {Manuel Kauers and Veronika Pillwein and Susanne Saminger-Platz},

title = {{Dominance in the Family of Sugeno-Weber t-norms}},

language = {english},

abstract = {The dominance relationship between two members of the family of Sugeno Weber t-norms is proven by using a quantifer elimination algorithm. Further it is shown that dominance is a transitive, and therefore also an order relation, on this family of t-norms.},

journal = {Fuzzy Sets and Systems},

volume = {181},

number = {1},

pages = {74--87},

isbn_issn = {ISSN 0165-0114},

year = {2011},

month = {October},

refereed = {yes},

length = {14}

}

**article**{RISC4328,author = {Manuel Kauers and Veronika Pillwein and Susanne Saminger-Platz},

title = {{Dominance in the Family of Sugeno-Weber t-norms}},

language = {english},

abstract = {The dominance relationship between two members of the family of Sugeno Weber t-norms is proven by using a quantifer elimination algorithm. Further it is shown that dominance is a transitive, and therefore also an order relation, on this family of t-norms.},

journal = {Fuzzy Sets and Systems},

volume = {181},

number = {1},

pages = {74--87},

isbn_issn = {ISSN 0165-0114},

year = {2011},

month = {October},

refereed = {yes},

length = {14}

}

### On the Average Complexity for the Verification of Compatible Sequences

#### C. Koukouvinos, V. Pillwein, D.E. Simos, Z. Zafeirakopoulos

Information Processing Letters 111(17), pp. 825-830. 2011. [pdf]@

author = {C. Koukouvinos and V. Pillwein and D.E. Simos and Z. Zafeirakopoulos},

title = {{On the Average Complexity for the Verification of Compatible Sequences }},

language = {english},

journal = {Information Processing Letters },

volume = {111},

number = {17},

pages = {825--830},

isbn_issn = {?},

year = {2011},

refereed = {yes},

length = {6}

}

**article**{RISC4352,author = {C. Koukouvinos and V. Pillwein and D.E. Simos and Z. Zafeirakopoulos},

title = {{On the Average Complexity for the Verification of Compatible Sequences }},

language = {english},

journal = {Information Processing Letters },

volume = {111},

number = {17},

pages = {825--830},

isbn_issn = {?},

year = {2011},

refereed = {yes},

length = {6}

}